Non-smooth Function Of The First Order Second-order Generalized Derivative

First-order and second-order generalized derivatives of nonsmooth function are themain parts of nonsmooth analysis.On convex analysis,we discuss convex function,Lipschitz function,lower semi-continuity and so on.Without smooth property,theclassical non-line programming theory and algorithm based on di?erentiable concept.Now,di?erentiable concept is generalized,corresponding optimal theory and compute methodare built on generalized derivative theory.Now,first-order generalized derivative is notsingle-valued mapping,but set-valued mapping.The main result gained on the paperare as follows :In the first chapter,the paper introduces first-order and second-order derivatives ofsmooth function and the research background on generalized function corresponding tonon-smooth function.At the same time,main results are introduced.In the second chapter,the paper gets the result that convex function is a kind ofimportant nonsmooth function,When convex property generalized first-order deriva-tive is studied,the paper draws into the notion of first-order di?erence quotiont func-tions and outer semi-continuity to get converge uniformly of its di?erence quotiont func-tions.Then,the paper combines Rademacher theorem to get the result that functionf(x) is almost di?erentiable in every place.Now,first-order di?erence of convex func-tion almost exists in every place.When its generalized second-order di?erentiable isstudied,the paper draws into second-order di?erence quotiont functions of ?f and useAlexandrov's theorem and Mignot theorem to get converge uniformly of two di?erencequotiont functions and their relation.In the third chapter,the notion of direction derivative is generalized to the local Lip-schitz function and lower semicontinuity function ,the notion of sub-di?erentiable of thelocal Lispchitz function and approximate subdi?erentiable of lower semicontinuity,andthe relation of first-order generalized derivative is studied.On these basis ,the studyin convex function based on semi-derivative is developped.